Stability and Super Stability of Fuzzy Approximately Ring Homomorphisms and Fuzzy Approximately Ring Derivations

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 2, Pages: 582-588

J. Environ. Treat. Tech.

ISSN: 2309-1185

Journal web link: http://www.jett.dormaj.com

Stability and Super Stability of Fuzzy Approximately

Ring Homomorphisms and Fuzzy Approximately

Ring Derivations

N. Eghbali

Department of Mathematics, Facualty of Mathematical Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran

Received: 05/08/2014

Accepted: 15/02/2020

Published: 20/05/2020

Abstract

In this paper, we establish the Hyers-Ulam-Rassias stability of ring homomorphisms and ring derivations in the uniform case on fuzzy

Banach algebras.

Keywords: Fuzzy normed space; Approximately ring homomorphism; Stability

Introduction¹

It seems that the stability problem of functional equations had

퐷(푎+푏)=퐷(푎)+퐷(푏),

퐷(푎푏)=퐷(푎)푏+퐷(푏)푎,

for every 푎,푏∈퐴;

for every 푎,푏∈퐴.

been first raised by Ulam [12]. An answer to this problem has

been given at first by Hyers [5] and then by Th. M. Rassias as

follows [10]. Suppose 퐸 and 퐸 are two real Banach spaces and

푓:퐸 →퐸 is a mapping. If there exist 훿≥0and 0≤푝<ꢀsuch

that ||푓(푥+푦)−푓(푥)−푓(푦)||≤훿(||푥|| +||푦|| ) for all

푥,푦∈퐸, then there is a unique additive mapping 푇:퐸 →퐸

It is of interest to consider an approximately ring derivation

on a Banach algebra. First of all, does there exist an

approximately ring derivation 푓 which is not an exact ring

derivation? If such a mapping 푓do exist, then it seems natural to

consider the following stability problem: does there exist ring

derivation near to 푓? The purpose of this paper is to prove the

stability of fuzzy approximately ring derivations. In fact, under a

mild assumption that 퐴is without order, we show the Bourgin-

type [1] super stability result.

ꢁ

such that ||푓(푥)−푇(푥)||≤ꢂ훿||푥|| /|ꢂ−ꢂ |for every 푥∈퐸.

In 1991, Gajda [2] gave a solution to this question for 푝>ꢀ. For

the case 푝=ꢀ, Th. M. Rassias and Šemrl [11] showed that there

exists a continuous real-valued function 푓:ℝ→ℝsuch that 푓can

not be approximated with an additive map. In 1992, Gavruta [3]

generalized the result of Rassias for the admissible control

functions. Moreover the approximately mappings have been

studied extensively in several papers. (See for instance [6], [7]).

Fuzzy notion introduced firstly by Zadeh [13] that has been

widely involved in different subjects of mathematics. Zadeh’s

definition of a fuzzy set characterized by a function from a

nonempty set 푋to [0,ꢀ]. Later, in 1984 Katsaras [8] defined a

fuzzy norm on a linear space to construct a fuzzy vector

topological structure on the space. Defining the class of

approximately solutions of a given functional equation one can

ask whether every mapping from this class can be somehow

approximated by an exact solution of the considered equation in

the fuzzy Banach algebra.

2 Preliminaries

In this section, we provide a collection of definitions and

related results which are essential and used in the next

discussions. Definition 2.1 Let 푋 be a real linear space. A

function 푁:푋×ℝ→[0,ꢀ]is said to be a fuzzy norm on 푋if for

all 푥,푦∈푋and all 푡,푠∈ℝ,

(N1) 푁(푥,푐)=0for 푐≤0;

(N2) 푥=0if and only if 푁(푥,푐)=ꢀfor all 푐>0;

ꢃ

(N3) 푁(푐푥,푡)=푁(푥, )if 푐≠0;

|ꢄ|

(N4) 푁(푥+푦,푠+푡)≥푚푖푛{푁(푥,푠),푁(푦,푡)};

(N5) 푁(푥,.) is non-decreasing function on ℝ and

푙푖푚 푁(푥,푡)=ꢀ;

ꢃ→∞

To answer this question, we use here the definition of fuzzy

normed spaces given in [8] to exhibit some reasonable notions of

fuzzy approximately ring homomorphism in fuzzy normed

algebras and we will prove that under some suitable conditions an

approximately ring homomorphism 푓from an algebra 푋into a

fuzzy Banach algebra 푌can be approximated in a fuzzy sense by

a ring homomorphism 푇from 푋to 푌. Let 퐴be a real or complex

Banach algebra. A mapping 퐷:퐴→퐴 is said to be a ring

derivative if

(N6) for 푥≠0, 푁(푥,.)is (upper semi) continuous on ℝ.

The pair (푋,푁)is called a fuzzy normed linear space.

Example 2.2 Let (푋,||.||)be a normed linear space. Then

Corresponding author: N. Eghbali, Department of Mathematics, Facualty of Mathematical Sciences, University of Mohaghegh Ardabili,

6199-11367, Ardabil, Iran. E-mail: nasrineghbali@gmail.com,eghbali@uma.ac.ir.

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 2, Pages: 582-588

푡

ꢈ푡≤ꢈꢈ0;ꢈ

∞

ꢊꢏꢋ

ꢊ ꢐꢊ ꢐꢊ

ꢂ 휑(ꢂ 푥,ꢂ 푦)<ꢎ.

∑

ꢇ

ꢈ0<푡≤ꢈꢈꢉ|푥|ꢉ;ꢈ

푁(푥,푡)=

ꢔ

ꢉ|푥|ꢉ

In particular, the similar results hold for 휑(푥,푦)=||푥|| +

ꢆ

ꢅ

ꢔ

|푦|| , where 푞>ꢀ. Stability and super stability of fuzzy

ꢀ

ꢈ푡>ꢉ|푥|ꢉ.

approximately ring homomorphisms and fuzzy approximately

ring derivations (N. Eghbali Tue Dec 24 00:16:25 2019).

is a fuzzy norm on 푋. Definition 2.3 Let (푋,푁)be a fuzzy

normed linear space and {푥 }be a sequence in 푋. Then {푥 }is

Stability of fuzzy approximately ring

homomorphism

ꢊ

said to be convergent if there exists 푥∈푋 such that

푙푖푚 푁(푥 −푥,푡)=ꢀfor all 푡>0. In that case, 푥is called

We start our work with definition of fuzzy approximately ring

ꢊ→∞

ꢊ

homomorphism. Definition 3.1 Let 푋be a linear algebra, (푌,푁)

a fuzzy Banach algebra and 휃≥0. We say that 푓:푋→푌is a

fuzzy approximately ring homomorphism map if

the limit of the sequence {푥 } and we denote it by 푁−

ꢊ

푙푖푚 푥 =푥. Definition 2.4 A sequence {푥 }in 푋 is called

ꢊ→∞ ꢊ

ꢊ

Cauchy if for each 휀>0and each 푡>0there exists 푛 such that

ꢋ

for all 푛≥푛 and all 푝>0, we have 푁(푥 −푥 ,푡)>ꢀ−휀.

ꢔ

ꢋ

ꢊꢌꢁ

ꢊ

푙푖푚 푁(푓(푥푦)−푓(푥)푓(푦),푡휃||푥|| ||푦|| )=ꢀ,

ꢃ→∞

It is known that every convergent sequence in a fuzzy normed

space is Cauchy and if each Cauchy sequence is convergent, then

the fuzzy norm is said to be complete and furthermore the fuzzy

normed space is called a fuzzy Banach space. Let 푋be an algebra

and (푋,푁)be complete fuzzy normed space. The pair (푋,푁)is

said to be a fuzzy Banach algebra if for every 푥,푦∈푋and 푠,푡∈

ℝ we have 푁(푥푦,푠푡)≥푚푖푛{푁(푥,푠),푁(푦,푡)}. Example 2.5

Let(푋,||.||)be a Banach algebra. Define,

uniformly on 푋×푋. Theorem 3.2 Let 푋 be a normed linear

algebra and (푌,푁)a fuzzy Banach algebra. Let 휃≥0and 푞≥

,푞≠ꢀ. Suppose that 푓:푋→푌is a function such that

ꢔ

푙푖푚 푁(푓(푥+푦)−푓(푥)−푓(푦),푡휃(||푥|| +||푦|| ))=ꢀ,

ꢃ→∞

uniformly on 푋×푋and

ꢔ

푙푖푚 푁(푓(푥푦)−푓(푥)푓(푦),푡휃||푥|| ||푦|| )=ꢀ,

(3.1)

ꢃ→∞

ꢀ

, ꢈ푎≤ꢈꢈ||푥||;ꢈ

, ꢈ푎>||푥||.

푁(푥,푎)=ꢍ

uniformly on 푋×푋. Then there is a unique ring homomorphism

푇:푋→푌such that

Then (푋,푁)is a fuzzy Banach algebra. Theorem 2.6 Let 푋

be a linear space and (푌,푁)be a fuzzy Banach space. Let 휑:푋×

푋→[0,ꢎ)be a control function such that

2ꢕꢃ||ꢓ||^ꢖ

푙푖푚 푁(푇(푥)−푓(푥), ₁_ꢐ₂_ꢖ_ꢗ_ꢘ_|)=ꢀ

ꢃ→∞

∞

ꢐꢊ

ꢊ

휑̃ (푥,푦)=∑ ꢂ 휑(ꢂ 푥,ꢂ 푦)<ꢎ,

ꢊꢏꢋ

uniformly on 푋. Proof. Theorem 2.6 and Corollary 2.7 show that

there exists a unique additive mapping 푇such that

for all 푥,푦∈푋. Let 푓:푋→푌 be a uniformly approximately

additive function with respect to 휑in the sense that

ꢕꢃ||ꢓ||^ꢖ

|⁾⁼^ꢀ^,

_|₁_ꢐ₂ꢖꢗꢘ

푙푖푚 푁(푇(푥)−푓(푥),

(3.2)

푙푖푚 푁(푓(푥+푦)−푓(푥)−푓(푦),푡휑(푥,푦))=ꢀ

ꢃ→∞

ꢒ

ꢑ(2 ꢓ)

₂ꢒ

where 푥∈푋. Now we only need to show that 푇is a multiplicative

uniformly on 푋×푋. Then 푇(푥)=푁−푙푖푚_ꢊ_→_∞

for all 푥∈

1ꢐꢔ

function. Put 푠=

and fix 푥,푦∈푋arbitrarily. By (3.2) we

푋exists and defines an additive mapping 푇:푋→푌such that if

for some 훿>0, 훼>0

|1ꢐꢔ|

have

ꢗꢚ

ꢚ

ꢖ

ꢕꢃꢊ ||ꢊ ꢓ||

푁(푓(푥+푦)−푓(푥)−푓(푦),훿휑(푥,푦))>훼,

for all 푥,푦∈푋; then

ꢐꢙ

ꢙ

ꢐꢙ

ꢙ

푙푖푚 푁(푛 푇(푛 푥)−푛 푓(푛 푥),

)=ꢀ.

ꢃ→∞

₁_ꢐ₂ꢖꢗꢘ

Using the additivity of 푇we have:

푁(푇(푥)−푓(푥),훿/ꢂ 휑̃ (푥,푥))>훼,

ꢕꢃꢊ^ꢚ⁽^ꢖ^ꢗ^ꢘ⁾||ꢓ||^ꢖ

)=ꢀ.

ꢐꢙ

ꢙ

푙푖푚 푁(푇(푥)−푛 푓(푛 푥),

ꢃ→∞

|1ꢐ2^ꢖ^ꢗ^ꢘ

for every 푥∈푋. Proof. [9]. Corollary 2.7 Let 푋be a normed

linear space and (푌,푁)a fuzzy Banach space. Let 휃≥0and 0≤

푞<ꢀ. Suppose that 푓:푋→푌is a function such that

ꢕꢃꢊ^ꢚ⁽^ꢖ^ꢗ^ꢘ⁾||ꢓ||^ꢖ

^Sⁱⁿ^c^e^푠⁽^푞⁻^ꢀ⁾^<⁰^w^e^o^b^t^aⁱⁿ^푙^푖^푚ꢊ→∞

=0. So

ꢔ

₁_ꢐ₂ꢖꢗꢘ

푙푖푚 푁(푓(푥+푦)−푓(푥)−푓(푦),푡휃(||푥|| +||푦|| ))=ꢀ

ꢃ→∞

ꢕꢃꢊ^ꢚ⁽^ꢖ^ꢗ^ꢘ⁾||ꢓ||^ꢖ

there is some 푛 >0such that

<푡for all 푛≥푛₁

₁_ꢐ₂ꢖꢗꢘ

uniformly on 푋×푋. Then there is a unique additive mapping

푇:푋→푌such that

ꢐꢙ ꢙ

Hence 푙푖푚 푁(푇(푥)−푛 푓(푛 푥),푡)≥

ꢃ→∞

ꢚ(ꢖꢗꢘ) ꢖ

2ꢕꢃꢊ ||ꢓ||

|1ꢐ2^ꢖ^ꢗ^ꢘ

and 푡>0.

푙푖푚 푁(푇(푥)−푛 푓(푛 푥),

ꢐꢙ

ꢙ

). So

ꢃ→∞

_ꢕ_ꢃ_|_|_ꢓ_|_|ꢖ

)=ꢀ

푙푖푚 푁(푇(푥)−푓(푥),

ꢃ→∞

_ꢐ₂ꢖꢗꢘ

ꢐꢙ

ꢙ

푁−푙푖푚 푛 푓(푛 푥)=푇(푥)ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ(3.3)

ꢊ→∞

uniformly on 푋. Proof. [9].

ꢊ

ꢐꢊ

Remark 2.8 Using the sequence {ꢂ 푓(ꢂ 푥)}, one can get

dual version of Theorem 2.6 and Corollary 2.7 when the control

function satisfies

Using (3.1) we get

ꢙ

ꢔ

푙푖푚 푁(푓((푛 푥)푦)−푓(푛 푥)푓(푦),푡휃||푛 푥|| ||푦|| )=ꢀ,

ꢃ→∞

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 2, Pages: 582-588

for all 푥,푦∈푋. Thus

Moreover for each fuzzy norm 푁on 푋, we have

ꢐꢙ

ꢙ

푙푖푚 푁(푛 푓((푛 푥)푦)−

푁(푓(푥+푦)−푓(푥)−푓(푦),푡(||푥||+||푦||))

ꢃ→∞

ꢐꢙ

ꢙ

ꢙ(ꢔꢐ1)

ꢔ

푛 푓(푛 푥)푓(푦),푡휃푛

||푥|| ||푦|| )=ꢀ,

=푁(훽푥 (||푥+푦||−||푥||−||푦||),푡(||푥||+||푦||))

ꢋ

ꢃ(||ꢓ||ꢌ||ꢝ||)

ꢙ(ꢔꢐ1)

ꢔ

=푁(훽푥_ꢋ,

)≥푁(훽푥 ,푡)(푥,푦∈푋,푡∈ℝ).

ꢋ

||ꢓꢌꢝ||ꢐ|ꢓ||ꢐ||ꢝ||

for all 푥,푦∈푋. Since 푙푖푚 푡휃푛

||푥|| ||푦|| =0, there is

ꢊ→∞

some 푛 >0such that

Therefore by the item (N5) of the Definition 2.1, we get

푡휃푛^ꢙ⁽^ꢔ^ꢐ¹⁾||푥|| ||푦|| <푡,

ꢔ

푙푖푚 푁(푓(푥+푦)−푓(푥)−푓(푦),푡(||푥||+||푦||))=ꢀ,

ꢃ→∞

uniformly on 푋×푋. Also,

for all 푛≥푛 . We get

푁(푓(푥푦)−푓(푥)푓(푦),푡||푥||||푦||)=푁(훼푥푦+훽푥 ||푥푦||−

ꢋ

ꢐꢙ

ꢙ

ꢐꢙ

ꢙ

푙푖푚 푁(푛 푓((푛 푥)푦)−푛 푓(푛 푥)푓(푦),푡)≥

ꢃ→∞

ꢐꢙ

ꢙ

(훼푥+훽푥 ||푥||)(훼푦+훽푥 ||푦||),푡||푥||||푦||)=푁(훼푥푦+

ꢋ

푙푖푚 푁(푛 푓((푛 푥)푦)−

ꢃ→∞

ꢐꢙ

ꢙ

ꢙ(ꢔꢐ1)

ꢔ

푛 푓(푛 푥)푓(푦),푡휃푛

||푥|| ||푦|| ).

훽푥 ||푥푦||−훼 푥푦−훼훽푥푥 ||푦||−훼훽푥 푦||푥||−

ꢋ

훽 푥 ||푥||||푦||,푡||푥||||푦||)≥푚푖푛{푁((ꢀ−

ꢋ

푁−푙푖푚 푛 푓ꢛ(푛 푥)푦ꢜ=푁−

ꢃ||ꢓ||||ꢝ||

ꢐꢙ

ꢙ

2 2

ꢋ

훼)훼푥푦,

),푁(||푥푦||훽푥_ꢋ,

),푁(훽 푥 ||푥||||푦||,

ꢊ→∞

ꢐꢙ

ꢙ

푙푖푚 푛 푓(푛 푥)푓(푦)ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ(3.4)

ꢊ→∞

ꢃ||ꢓ|||ꢝ||

ꢃ||ꢓ||||ꢝ||

)}

푁(훼훽푥푥 ||푦||,

),푁(훼훽푥 푦||푥||,

By putting 푛 =푚푖푛{푛 ,푛 }and applying (3.3) and (3.4) we

ꢋ

1 2

have

ꢐꢙ

ꢙ

푇(푥푦)=푁−푙푖푚 푛 푓(푛 (푥푦))=푁−

ꢙ

ꢊ→∞

where 푥∈푋and 푡∈ℝ. Taking into account the following

inequalities

ꢐꢙ

푙푖푚 푛 푓(푛 푥)푓(푦)=푇(푥)푓(푦),

ꢊ→∞

for all 푥,푦∈푋. From this equation by the additivity of 푇we have:

ꢃꢉ|ꢓ|ꢉꢉ|ꢝ|ꢉ

ꢟ=푁ꢠ훼푥푦, ₅_|₁_ꢐ_ꢡ_|ꢢ≥

ꢙ

푁ꢞ(ꢀ−훼)훼푥푦,

푇(푥)푓(푛 푦)=푇(푥(푛 푦))=푇(푛 푥)푓(푦)=푛 푇(푥)푓(푦),

ꢐꢙ

ꢙ

for all 푥,푦∈푋. Therefore 푇(푥)푛 푓(푛 푦)=푇(푥)푓(푦). By

letting 푛tend to infinity we see that 푇(푥)푇(푦)=푇(푥)푓(푦)for

all 푥,푦∈푋. To prove the uniqueness property of 푇, assume that

ꢃ

푁ꢠ훼푥푦, ꢢꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ(3.5)

∗

푇 is another ring homomorphism satisfying

ꢃ||ꢓ||||ꢝ||

)≥푁(훽푥 ,푡/ꢣ), (3.6)

ꢋ

5||ꢓꢝ||

푁(||푥푦||훽푥_ꢋ,

)=푁(훽푥_ꢋ,

ꢂ휃푡||푥||^ꢔ

∗

푙푖푚 푁(푇 (푥)−푓(푥),

)=ꢀ

ꢔꢐ1

ꢃ→∞

ꢀ−ꢂ |

ꢃ||ꢓ||||ꢝ||

ꢋ

2 2

푁(훽 푥 ||푥||||푦||,

)=푁(훽 푥 ,푡/ꢣ),

ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ(3.7)

ꢋ

We have

ꢃ||ꢓ||||ꢝ||

ꢃ||ꢓ||

ꢋ 5|ꢤ|

∗

ꢐꢙ

ꢙ

푁(훼훽푥푥 ||푦||,

)=푁(훼푥푥 , )≥푁(훼푥푥 ,푡/ꢣ),(3.8)

푁(푇(푥)−푇 (푥),푡)≥푚푖푛{푁(푇(푥)−푛 푓(푛 푥),푡/

ꢋ

∗

ꢐꢙ

ꢙ

ꢂ),푁(푇 (푥)−푛 푓(푛 푥),푡/ꢂ)}

ꢃ||ꢓ||||ꢝ||

ꢃ||ꢝ||

푁(훼훽푥 푦||푥||,

)=푁(훼푥 푦, )≥푁(훼푥 푦,푡/ꢣ),(3.9)

ꢋ ꢋ

5|ꢤ|

Given, 휀>0by (3.3) we can find some 푡 >0such that

ꢋ

ꢐꢙ

ꢙ

it can be easily seen that 푙푖푚 푁(푓(푥푦)−

푁(푇(푥)−푛 푓(푛 푥),푡/ꢂ)≥ꢀ−휀,

ꢃ→∞

푓(푥)푓(푦),푡||푥||||푦||)=ꢀuniformly on 푋×푋and therefore the

conditions of Theorem 3.2 are fulfilled. Now we suppose that

there exists a unique ring homomorphism 푇 satisfying the

conditions of Theorem 3.2. By the equation

and

∗

ꢐꢙ

ꢙ

푁(푇 (푥)−푛 푓(푛 푥),푡/ꢂ)≥ꢀ−휀,

푙푖푚 푁ꢠ푓(푥+푦)−푓(푥)−푓(푦),푡ꢛꢉ|푥|ꢉ+ꢉ|푦|ꢉꢜꢢ=ꢀ(3.10)

for all 푡/ꢂ≥푡 , 푥∈푋and 푛∈ℕ. So

ꢃ→∞

ꢋ

∗

for given 휀>0, we can find some 푡 >0such that

푁(푇(푥)−푇 (푥),푡)≥ꢀ−휀,

ꢋ

푁(푓(푥+푦)−푓(푥)−푓(푦),푡(||푥||+||푦||))≥ꢀ−휀,

for all 푡>0. Hence by items (N5) and (N2) of definition 2.1 we

have 푇(푥)=푇 (푥)for all 푥∈푋. In the following example we

will show that Theorem 3.2 does not necessarily hold for 푞=ꢀ.

∗

for all 푥,푦∈푋and all 푡≥푡 . By using the simple induction on

ꢋ

푛, we shall show that

Example 3.3 Let 푋be a Banach algebra, 푥 ∈푋and 훼,훽are

real numbers such that |훼|≥ꢀ−||푥||||푦||and |훽|≤||푥||for

every 푥∈푋. Put:

ꢋ

ꢊ

푁(푓(ꢂ 푥)−ꢂ 푓(푥),푡푛ꢂ ||푥||)≥ꢀ−휀.

ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ(3.11)

putting 푦=푥in (3.10), we get (3.11) for 푛=ꢀ. Let (3.11) holds

for some positive integer 푛. Then

푓(푥)=훼푥+훽푥 ||푥||,(푥∈푋).

ꢋ

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 2, Pages: 582-588

ꢊꢌ1

2ꢕꢃꢊ^ꢚ⁽^ꢖ^ꢗ^ꢘ⁾||ꢓ||^ꢖ

ꢙ

푁(푓(ꢂ 푥)−ꢂ 푓(푥),푡(푛+ꢀ)ꢂ ||푥||)≥

ꢐꢙ

ꢙ

ꢐꢙ

ꢊꢌ1

ꢊ

푙푖푚 푁(푛 퐷(푛 푥)−푛 푓(푛 푥),

)=ꢀ,

ꢃ→∞

|1ꢐ2^ꢖ^ꢗ^ꢘ

푚푖푛{푁(푓(ꢂ 푥)−ꢂ푓(ꢂ 푥),푡(||ꢂ 푥||+

ꢊ

ꢊꢌ1

ꢊꢐ1

|ꢂ 푥||)),푁(ꢂ푓(ꢂ 푥)−ꢂ 푓(푥),ꢂ푡푛(||ꢂ 푥||+

ꢊꢐ1

|ꢂ 푥||))≥ꢀ−휀.

for all 푥∈푋. By the additivity of 퐷,

ꢕꢃꢊ^ꢚ⁽^ꢖ^ꢗ^ꢘ⁾||ꢓ||^ꢖ

This completes the induction argument. We observe that

ꢐꢙ

ꢙ

푙푖푚 푁(퐷(푥)−푛 푓(푛 푥),

)=ꢀ,

ꢃ→∞

₁_ꢐ₂ꢖꢗꢘ

푙푖푚 푁(푇(푥)−푓(푥),푛푡||푥||)≥ꢀ−휀.

ꢊ→∞

for

all 푥∈푋.Since 푠(푞−ꢀ)<0

obtain

ꢚ(ꢖꢗꢘ) ꢖ

||ꢓ||

=0. So there is some 푛 >0 such that

Hence

2ꢕꢃꢊ

푙푖푚_ꢊ_→_∞

₁_ꢐ₂ꢖꢗꢘ

2ꢕꢃꢊ^ꢚ⁽^ꢖ^ꢗ^ꢘ⁾||ꢓ||^ꢖ

푙푖푚 푁ꢛ푇(푥)−푓(푥),푛푡ꢉ|푥|ꢉꢜ=ꢀꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ

(3.12)

ꢊ→∞

푡

for all 푛≥푛₁and 푡>0. Hence

1ꢐ2^ꢖ^ꢗ^ꢘ

ꢐꢙ ꢙ

푙푖푚ꢃ→∞푁(푇(푥)−푛 푓(푛 푥),푡)≥푙푖푚ꢃ→∞푁(푇(푥)−

ꢚ(ꢖꢗꢘ) ꢖ

One may regard 푁(푥,푡)as the truth value of the statement

the norm of 푥is less than or equal to the real number 푡’. So (3.12)

is a contradiction with the non-fuzzy sense. This means that there

’

ꢐꢙ

ꢙ

2ꢕꢃꢊ

||ꢓ||

). So

푛 푓(푛 푥),

|1ꢐ2^ꢖ^ꢗ^ꢘ

is no such a 푇.

ꢐꢙ

ꢙ

푁−푙푖푚 푛 푓(푛 푥)=퐷(푥).

ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ(4.3)

ꢊ→∞

Stability and super stability of fuzzy

approximately ring derivation

By (4.1) we have

We start our work with definition of fuzzy approximately ring

derivation. Definition 4.1 Let (푋,푁)be a fuzzy Banach algebra

and 휃≥0. We say that 푓:푋→푋is a fuzzy approximately ring

derivation if

ꢐꢙ

ꢙ

푙푖푚 푁(푛 푓((푛 푥)푦)−푥푓(푦)−

ꢃ→∞

ꢔ

ꢐꢙ

ꢙ

ꢐꢙ

ꢙ

ꢔ

푛 푓(푛 푥)푦,푡휃푛 ||푛 푥|| ||푦|| )=ꢀ,

ꢔ

for all 푥,푦∈푋. Since 푠(푞−ꢀ)<0 we obtain

푙푖푚 푁(푓(푥푦)−푥푓(푦)−푓(푥)푦,푡휃||푥|| ||푦|| )=ꢀ,

ꢃ→∞

ꢙ(ꢔꢐ1)

ꢔ ꢔ

||푥|| ||푦|| =0. So there is some 푛 >0such

ꢔ ꢔ

푙푖푚 푡휃푛

ꢊ→∞

that 푡휃푛

ꢙ(ꢔꢐ1)

||푥|| ||푦|| <푡, for all 푛≥푛 and 푡>0. Hence:

uniformly on 푋×푋. Definition 4.2 Let (푋,푁)be a fuzzy Banach

a fuzzy

algebra and 휃≥0. We say that 푓:푋→푋 is

ꢐꢙ

ꢙ

ꢐꢙ

ꢙ

푙푖푚 푁(푛 푓((푛 푥)푦)−푥푓(푦)−푛 푓(푛 푥)푦,푡)≥

ꢃ→∞

approximately Jordan derivation if

ꢙ

푙푖푚 푁(푛 푓((푛 푥)푦)−푥푓(푦)−

ꢃ→∞

ꢐꢙ

ꢙ

ꢙ(ꢔꢐ1) ꢔ ꢔ

||푥|| ||푦|| ).

2ꢔ

푛 푓(푛 푥)푦,푡휃푛

푙푖푚 푁(푓(푥 )−ꢂ푥푓(푥),푡휃||푥|| )=ꢀ,

ꢃ→∞

uniformly on 푋×푋. Theorem 4.3 Let (푋,푁)be a fuzzy Banach

algebra, 휃≥0and 푞≥0,푞≠ꢀ. Suppose that 푓:푋→푋is a

function such that

ꢐꢙ

ꢙ

푁−푙푖푚 푛 푓((푛 푥)푦)=푁−푙푖푚 (푥푓(푦)+

ꢊ→∞

ꢐꢙ

ꢙ

푛 푓(푛 푥)푦)ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ

(4.4)

ꢔ

푙푖푚 푁(푓(푥+푦)−푓(푥)−푓(푦),푡휃(||푥|| +||푦|| ))=ꢀ,

ꢃ→∞

By putting 푛 =푚푖푛{푛 ,푛 }and applying (4.3) and (4.4 we get

ꢋ

1 2

uniformly on 푋×푋and

ꢐꢙ

ꢙ

퐷(푥푦)=푁−푙푖푚 푛 푓(푛 (푥푦))=푁−

ꢊ→∞

ꢐꢙ

ꢙ

ꢔ

푙푖푚ꢊ→∞푛 푓((푛 푥)푦)ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ

(4.5)

all

푙푖푚 푁(푓(푥푦)−푥푓(푦)−푓(푥)푦,푡휃||푥|| ||푦|| )=ꢀ, (4.1)

ꢃ→∞

ꢐꢙ

ꢙ

푥푓(푦)+푁−푙푖푚 푛 푓(푛 푥)푦=푥푓(푦)+퐷(푥)푦,ꢈfor

ꢊ→∞

uniformly on 푋×푋. Then there is a unique ring derivation

퐷:푋→푋such that

푥,푦∈푋. Then by (4.5) and the additivity of 퐷, we have

푥푓(푛 푦)+푛 퐷(푥)푦=푥푓(푛 푦)+퐷(푥)푛 푦=퐷(푥(푛 푦))=

퐷((푛 푥)푦)=푛 푥푓(푦)+퐷(푛 푥)푦=푛 푥푓(푦)+푛 퐷(푥)푦.

Therefore 푥푓(푦)=푥푛 푓(푛 푦) for all 푥,푦∈푋. By letting 푛

tend to infinity we see that 푥푓(푦)=푥퐷(푦) for all 푥,푦∈푋.

Combining this formula with equation (4.5) we get 퐷(푥푦)=

푥퐷(푦)+퐷(푥)푦 for all 푥,푦∈푋. The proof of uniqueness

property of 퐷is similar to the proof of Theorem 3.2.

Corollary 4.4 Theorem 4.3 satisfies for fuzzy approximately

Jordan derivation.

ꢙ

ꢕꢃ||ꢓ||^ꢖ

ꢐꢙ

ꢙ

푙푖푚 푁(퐷(푥)−푓(푥),_|

_|)=ꢀ,

ꢃ→∞

₁_ꢐ₂ꢖꢗꢘ

uniformly on 푋. Proof. Theorem 2.6 and Corollary 2.7 show that

there exists a unique additive mapping 퐷such that

_ꢕ_ꢃ_|_|_ꢓ_|_|ꢖ

_|)=ꢀ,ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ

푙푖푚 푁(퐷(푥)−푓(푥),_|

(4.2)

ꢃ→∞

₁_ꢐ₂ꢖꢗꢘ

where 푥∈푋. Now we only need to show that 퐷is a map such

Proof. As same as the proof of Theorem 4.3 we have

1ꢐꢔ

that 퐷(푥푦)=푥퐷(푦)−퐷(푥)푦for all 푥,푦∈푋. Put 푠=_|

and

ꢐꢙ

ꢙ

퐷(푥)=푁−푙푖푚 푛 푓(푛 푥). (4.6)

1ꢐꢔ|

ꢊ→∞

A quite similar argument to the proof of Theorem 4.3 shows that

fix 푥,푦∈푋arbitrarily. By (4.2) we have

ꢐ2ꢙ

2ꢙ

퐷(푥 )=푁−푙푖푚 푛 푓(푛 푥). By Definition (4.2) we

ꢊ→∞

have

ꢃ→∞

ꢀ,

ꢚ

ꢖ

ꢐ2ꢙ

2ꢙ 2

ꢐꢙ

ꢙ

ꢐ2ꢙ

ꢙ

2ꢔ

ꢕꢃ||ꢊ ꢓ||

푙푖푚 푁(푛 푓(푛 푥 )−ꢂ푛 푥푓(푛 푥),푡휃푛 ||푛 푥|| )=

ꢙ

푙푖푚 푁(퐷(푛 푥)−푓(푛 푥),

)=ꢀ,

ꢃ→∞

1ꢐ2^ꢖ^ꢗ^ꢘ

for all 푥∈푋.

Since ꢂ푠(푞−ꢀ)<0we obtain 푙푖푚 푡휃푛

2ꢙ(ꢔꢐ1)

2ꢔ

||푥|| =0.

ꢊ→∞

for all 푥∈푋. Thus

So there is some 푛 >0such that

ꢋ

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 2, Pages: 582-588

푡휃푛²^ꢙ⁽^ꢔ^ꢐ¹⁾||푥|| <푡,

2ꢔ

for all 푛≥푛 and 푡>0. Hence

ꢋ

ꢐ2ꢙ

2ꢙ 2

ꢐꢙ

ꢙ

ꢐ1

푙푖푚 푁(푛 푓(푛 푥 )−ꢂ푛 푥푓(푛 푥),푡)≥

푁(푟푓(푛 푥푎)−푟푛 푥푓(푎)−푟푓(푛 푥)푎,푡/4)≥

ꢃ→∞

2ꢙ 2

ꢐ2ꢙ

푙푖푚 푁(푛 푓(푛 푥 )−

(4.12)

ꢃ→∞

ꢙ

2ꢙ(ꢔꢐ1)

2ꢔ

ꢐ1

ꢂ푛 푥푓(푛 푥),푡휃푛

||푥|| ),

푁(푟푓(푛 푥푎)−푟푛 푥푓(푎)−

ꢐ1

ꢐꢔ

ꢔ

for all 푥∈푋. So

푟푓(푛 푥)푎,푡휃푟푛 ||푥|| ||푎|| ).

ꢐ2ꢙ

2ꢙ

ꢙ

푁−푙푖푚 푛 푓(푛 푥)=푁−푙푖푚 ꢂ푛 푥푓(푛 푥).

By using (4.9) and (4.8) for given 휀>0we can find some 푡 ,푡 >

ꢊ→∞

1 2

(4.7)

0such that

Now by using (4.6) and (4.7) we get

ꢗꢖ ꢖ

ꢥꢃꢕꢊ ||ꢓꢦ||

ꢢ≥ꢀ−휀,ꢈ(4.13)

ꢐ1

푁ꢠ퐷ꢛ(푛 푥)(푟푎)ꢜ−푟푓(푛 푥푎),

ꢐ2ꢙ

2ꢙ 2

|1ꢐ2^ꢖ^ꢗ^ꢘ

퐷(푥 )=푁−푙푖푚 푛 푓(푛 푥 )=푁−

푙푖푚 ꢂ푛 푥푓(푛 푥)=ꢂ푥퐷(푥),

ꢊ→∞

ꢐꢙ

ꢙ

ꢊ→∞

for every 푥∈푋.

and

ꢐ1

푁(푟푓(푛 푥푎)−푟푛 푥푓(푎)−

ꢐ1

ꢐꢔ

ꢔ

푟푓(푛 푥)푎,푡휃푟푛 ||푥|| ||푎|| )≥ꢀ−휀,

Theorem 4.5 Let (푋,푁)be a fuzzy Banach algebra, 휃≥0and

푞≥0,푞≠ꢀ. Suppose that 푓:푋→푋is a function such that

for all 푡≥푡 . So by combining (4.10), (4.11), (4.12) and (4.13)

ꢔ

ꢋ

푙푖푚 푁(푓(푥+푦)−푓(푥)−푓(푦),푡휃(||푥|| +||푦|| ))=ꢀ,

ꢃ→∞

we get

uniformly on 푋×푋and

ꢔ

푙푖푚 푁(푓(푥푦)−푥푓(푦)−푓(푥)푦,푡휃||푥|| ||푦|| )=ꢀ,

ꢐ1

ꢃ→∞

푁(퐷((푛 푥)(푟푎))−푟푛 푥푓(푎)−푓(푛 푥)푟푎,푡/ꢂ)≥ꢀ−

휀. (4.14)

(4.8)

uniformly on 푋×푋.

Then we have

푥{푓(푟푎)−푟푓(푎)}=0,

for every 푎,푥∈푋and 푟∈ℚ−{0}.

By applying (4.8) we have

푙푖푚 푁(푛 푥푓(푟푎)+푓(푛 푥)푟푎−

ꢐ1

ꢊ→∞

ꢐꢔ ꢔ

ꢐ1

ꢔ

푓((푛 푥)(푟푎)),푡휃푛 푟 ||푥|| ||푎|| )=ꢀ.

ꢐꢔ ꢔ

ꢔ

Since 푙푖푚 푡휃푛 푟 ||푥|| ||푎|| =0, there exists 푛 >0

ꢊ→∞

ꢋ

such that

푡휃푛 푟 ||푥|| ||푎|| <푡/ꢂ

for every 푛≥푛 and 푡>0. So

Proof. Pick 푎,푥∈푋and 푟∈ℚ−{0}arbitrarily. By Theorem 4.3

there exists a unique ring derivation 퐷such that:

ꢐꢔ ꢔ

ꢔ

ꢋ

ꢕꢃ||ꢓ||^ꢖ

|⁾⁼^ꢀ^,

푙푖푚 푁(퐷(푥)−푓(푥),_|

(4.9)

ꢃ→∞

1ꢐ2^ꢖ^ꢗ^ꢘ

ꢐ1

푁(푛 푥푓(푟푎)+푓(푛 푥)푟푎−푓((푛 푥)(푟푎)),푡/ꢂ)≥

4.15)

(

ꢐ1 ꢐ1

푁(푛 푥푓(푟푎)+푓(푛 푥)푟푎−

ꢔ ꢔ

where 푥∈푋. Recall that 퐷is additive, and so it is easy to see

that 퐷(푟푏)=푟퐷(푏)for every 푏∈푋. Fix 푛∈ℕarbitrarily

and we get

ꢐ1

ꢐꢔ ꢔ

푓((푛 푥)(푟푎)),푡휃푛 푟 ||푥|| ||푎|| ),

for all 푡≥푡_ꢋ.

Given 휀>0, we can find some 푡ꢋ >0such that

ꢐ1

푁(퐷((푛 푥)(푟푎))−푟푛 푥푓(푎)−푓(푛 푥)푟푎,푡/ꢂ)≥

4.10)

(

ꢐ1

ꢐ1 ꢐ1

푁(푛 푥푓(푟푎)+푓(푛 푥)푟푎−

ꢐ1 ꢐꢔ ꢔ ꢔ

푚푖푛{푁(퐷((푛 푥)(푟푎))−푟푓(푛 푥푎),푡/4),푁(푟푓(푛 푥푎)−

푟푛 푥푓(푎)−푟푓(푛 푥)푎,푡/4)}.

ꢐ1

ꢔ

푓((푛 푥)(푟푎)),푡휃푛 푟 ||푥|| ||푎|| )≥ꢀ−휀,

4.16)

for all 푡≥푡 . So

(

By (4.9) we have

ꢋ

ꢗꢘ ꢖ

ꢥꢃꢕ||ꢊ ꢓꢦ||

)=ꢀ.

ꢐ1

푁(푛 푥푓(푟푎)+푓(푛 푥)푟푎−푓((푛 푥)(푟푎)),푡/ꢂ)≥ꢀ−

휀. (4.17)

푙푖푚 푁(퐷((푛 푥)(푟푎))−푟푓(푛 푥푎),

ꢊ→∞

₁_ꢐ₂ꢖꢗꢘ

ꢥꢃꢕ||ꢓꢦ||^ꢖ

=0, there is some 푛 >0 such that

Also we have:

Since 푙푖푚_ꢊ_→_∞

₁_ꢐ₂ꢖꢗꢘ

ꢗꢖ ꢖ

ꢃꢕꢊ ||ꢓꢦ||

푡/4for all 푛≥푛 and 푡>0. Hence

ꢐ1 ꢐ1 ꢐ1

푁(푓((푛 푥)(푟푎))−푟푛 푥푓(푎)−푓(푛 푥)푟푎,푡/ꢂ)≥

₁_ꢐ₂ꢖꢗꢘ

(

4.18)

ꢐ1

푚푖푛{푁(푓((푛 푥)(푟푎))−퐷((푛 푥)(푟푎)),푡/

ꢐ1

푁(퐷((푛 푥)(푟푎))−푟푓(푛 푥푎),푡/4)≥

4),푁(퐷((푛 푥)(푟푎))−푟푛 푥푓(푎)−푓(푛 푥)푟푎,푡/4)}.

ꢗꢖ ꢖ

ꢥꢃꢕꢊ ||ꢓꢦ||

ꢐ1

푁(퐷((푛 푥)(푟푎))−푟푓(푛 푥푎),

4.11)

1ꢐ2^ꢖ^ꢗ^ꢘ

(

By (4.9) we have

Also by (4.8) we get

ꢐ1

푙푖푚 푁(푟푓(푛 푥푎)−푟푛 푥푓(푎)−

ꢖ

ꢗꢖ

ꢖ

ꢊ→∞

ꢃꢕꢥ ꢊ ||ꢓꢦ||

ꢐ1

ꢔ

ꢐ1

푟푓(푛 푥)푎,푟푡휃||푛 푥|| ||푎|| )=ꢀ.

푙푖푚ꢊ→∞푁(푓((푛 푥)(푟푎))−퐷((푛 푥)(푟푎)),

1ꢐ2^ꢖ^ꢗ^ꢘ

ꢐꢔ

ꢔ

Since 푙푖푚 푡휃푟푛 ||푥|| ||푎|| =0, there is some 푛 >0such

ꢀ.

ꢊ→∞

that

ꢐꢔ

ꢔ

푡휃푟푛 ||푥|| ||푎|| <푡/4

Journal of Environmental Treatment Techniques

2020, Volume 8, Issue 2, Pages: 582-588

ꢗꢖ ꢖ ꢖ

ꢃꢕꢊ ꢥ ||ꢦꢓ||

=0, there is some 푛 >0such

2ꢕꢃꢊ^ꢚ⁽^ꢘ^ꢗ^ꢖ⁾||ꢓ||^ꢖ

Since 푙푖푚_ꢊ_→_∞

for every 푥∈푋. Since 푙푖푚_ꢊ_→_∞

=0, there is some

1ꢐ2^ꢖ^ꢗ^ꢘ

ꢋ

|1ꢐ2ꢖꢗꢘ|

ꢗꢖ

ꢖ

ꢃꢕꢊ ꢥ ||ꢦꢓ||

_ꢕ_ꢃ_ꢊꢚ(ꢘꢗꢖ)_|_|_ꢓ_|_|ꢖ

<푡for every 푛≥푛ꢋ. So

that

<푡/4, for all 푛≥푛 and 푡>0. Hence

ꢋ

₁_ꢐ₂ꢖꢗꢘ

푛ꢋ >0such that

₁_ꢐ₂ꢖꢗꢘ

| |

ꢐ1

푁(푓((푛 푥)(푟푎))−퐷((푛 푥)(푟푎)),푡/4)≥

ꢙ ꢐꢙ ꢙ ꢐꢙ ꢙ ꢐꢙ

푁(푛 퐷(푛 푥)−푛 푓(푛 푥),푡)≥푁(푛 퐷(푛 푥)−

ꢗꢖ ꢖ ꢖ

ꢃꢕꢊ ꢥ ||ꢦꢓ||

ꢐ1

푁(푓((푛 푥)(푟푎))−퐷((푛 푥)(푟푎)),

1ꢐ2^ꢖ^ꢗ^ꢘ

2ꢕꢃꢊ ||ꢊ ꢓ||

ꢚ

ꢗꢚ

ꢖ

ꢙ

ꢐꢙ

푛 푓(푛 푥),

1ꢐ2^ꢖ^ꢗ^ꢘ

By (4.9) for given 휀>0, we can find some 푡 >0such that

ꢋ

By using (4.23) for given 휀>0, we can find some 푡 >0

such that

ꢋ

ꢗꢖ ꢖ ꢖ

ꢃꢕꢊ ꢥ ||ꢦꢓ||

ꢢ≥ꢀ−

ꢐ1

푁ꢠ푓ꢛ(푛 푥)(푟푎)ꢜ−퐷ꢛ(푛 푥)(푟푎)ꢜ,

1ꢐ2^ꢖ^ꢗ^ꢘ

ꢚ

ꢗꢚ

ꢖ

ꢕꢃꢊ ||ꢊ ꢓ||

ꢙ

ꢐꢙ

ꢙ

ꢐꢙ

푁(푛 퐷(푛 푥)−푛 푓(푛 푥),

)≥ꢀ−휀,

휀,ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ(4.19)

₁_ꢐ₂ꢖꢗꢘ

for all 푡≥푡 . Hence

ꢋ

for all 푡≥푡 . Now by combining (4.14), (4.19) and (4.18) we get

ꢋ

푁(퐷(푥)−푓(푥),푡)=ꢀ,

ꢃ

ꢐ1

푁ꢠ푓ꢛ(푛 푥)(푟푎)ꢜ−푟푛 푥푓(푎)−푓(푛 푥)푟푎, ꢢ≥ꢀ−

휀.ꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈꢈ(4.20)

for all 푡>0. So by using item (N2) of Definition 2.1, we have

퐷(푥)=푓(푥) for every 푥∈푋, which shows that 푓 is a ring

derivation.

On the other hand, we have

ꢐ1

푁(푛 푥(푓(푟푎)−푟푓(푎)),푡)≥

(4.21)

Ethical issue

ꢐ1

Authors are aware of, and comply with, best practice in

publication ethics specifically with regard to authorship

푚푖푛{푁(푛 푥푓(푟푎)+푓(푛 푥)푟푎−푓((푛 푥)(푟푎)),푡/

ꢐ1

ꢂ),푁(푓((푛 푥)(푟푎))−푟푛 푥푓(푎)−푓(푛 푥)푟푎,푡/ꢂ)}.

(avoidance of guest authorship), dual submission, manipulation

So by combining (4.21), (4.20) and (4.17) we have

푁(푛 푥(푓(푟푎)−푟푓(푎)),푡)≥ꢀ−휀. Hence 푁(푥(푓(푟푎)−

of figures, competing interests and compliance with policies on

research ethics. Authors adhere to publication requirements that

submitted work is original and has not been published elsewhere

in any language.

ꢐ1

푟푓(푎)),푛푡)≥ꢀ−휀. Now by using item (N2) of Definition 2.1

we get 푥(푓(푟푎)−푟푓(푎))=0. In the following Theorem we

consider the conditions for super stability of approximately ring

derivations. Theorem 4.6 Let (푋,푁)be a fuzzy Banach algebra

without order, 휃≥0and 푞≥0,푞≠ꢀ. Suppose that 푓:푋→푋is

a function such that

Competing interests

The authors declare that there is no conflict of interest that

would prejudice the impartiality of this scientific work.

ꢔ

푙푖푚 푁(푓(푥+푦)−푓(푥)−푓(푦),푡휃(||푥|| +||푦|| ))=ꢀ,

ꢃ→∞

Authors’ contribution

All authors of this study have a complete contribution for data

collection, data analyses and manuscript writing

uniformly on 푋×푋and

ꢔ

푙푖푚 푁(푓(푥푦)−푥푓(푦)−푓(푥)푦,푡휃||푥|| ||푦|| )=ꢀ,

ꢃ→∞

(

4.22)

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1ꢐꢔ

ꢐꢙ

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ꢕꢃ||ꢓ||^ꢖ

_|)=ꢀ,

푙푖푚 푁(퐷(푥)−푓(푥),_|

ꢃ→∞

₁_ꢐ₂ꢖꢗꢘ

that

푙푖푚 푁ꢠ푛 퐷(푛 푥)−푛 푓(푛 푥),

ꢚ ꢗꢚ ꢖ

ꢕꢃꢊ ||ꢊ ꢓ||

ꢢ=ꢀ,ꢈꢈꢈ(4.23)

ꢙ

ꢐꢙ

ꢙ

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ꢃ→∞

₁_ꢐ₂ꢖꢗꢘ

29 .

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